Problem: Simplify and expand the following expression: $ \dfrac{4}{q - 7}+ \dfrac{5}{q - 3}+ \dfrac{q}{q^2 - 10q + 21} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{q}{q^2 - 10q + 21} = \dfrac{q}{(q - 7)(q - 3)}$ Now we have: $ \dfrac{4}{q - 7}+ \dfrac{5}{q - 3}+ \dfrac{q}{(q - 7)(q - 3)} $ The least common multiple of the denominators is: $ (q - 7)(q - 3)$ In order to get the first term over $(q - 7)(q - 3)$ , multiply by $\dfrac{q - 3}{q - 3}$ $ \dfrac{4}{q - 7} \times \dfrac{q - 3}{q - 3} = \dfrac{4(q - 3)}{(q - 7)(q - 3)} $ In order to get the second term over $(q - 7)(q - 3)$ , multiply by $\dfrac{q - 7}{q - 7}$ $ \dfrac{5}{q - 3} \times \dfrac{q - 7}{q - 7} = \dfrac{5(q - 7)}{(q - 7)(q - 3)} $ Now we have: $ \dfrac{4(q - 3)}{(q - 7)(q - 3)} + \dfrac{5(q - 7)}{(q - 7)(q - 3)} + \dfrac{q}{(q - 7)(q - 3)} $ $ = \dfrac{ 4(q - 3) + 5(q - 7) + q} {(q - 7)(q - 3)} $ Expand: $ = \dfrac{4q - 12 + 5q - 35 + q}{q^2 - 10q + 21} $ $ = \dfrac{10q - 47}{q^2 - 10q + 21}$